Consider a Polya urn model in which we have $N$ balls of which $k$ are black and the rest are white. We repeatedly draw balls (without replacement) in "batches" of size $m$ (assume $m$ divides $N$), and are interested in the probability that every batch contains fewer than $t$ black balls.

It is not too hard to write out an expression for the above probability. But I am looking for a closed-form expression, or at least nice upper/lower bounds on the probability in question. (Feel free to assume $N$ large if that helps.)

This seems like it should be a well studied question in reliability engineering, but I have not found any references. Can anyone help?